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The Numerical Calculation of Cusps, Bifurcation Points and Isola Formation Points in Two Parameter Problems

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Numerical Methods for Bifurcation Problems

Abstract

In this paper we discuss the numerical computation of solutions of the nonlinear, two parameter, problem

(1.1)

where x ∈ R n is a State variable, λ and α are parameters, and f is a smooth function. Many physical systems can be described by equations like (1.1), see for example, [l],[3],[l2] and [7],[17], where there are more than two parameters.

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References

  1. Bazley,N.W. and Wake, G.C. The disappearance of criticality in the theory of thermal ignition. ZAMP, 29, (1979), p. 971–976.

    Article  Google Scholar 

  2. Chan, T.F., On the existence and computation of LU-factorizations with small pivots (to appear in Maths. Comp.).

    Google Scholar 

  3. Cliffe, K.A., Numerical calculations of two-cell and Single cell Taylor flows. (To appear in J. Fluid Mech.).

    Google Scholar 

  4. Cliffe, K.A. and Spence, A., The Calculation of High Order Singularities in the Taylor Problem (this proceedings).

    Google Scholar 

  5. Dennis, J.E. and More, J.J., (1977) Quasi-Newton methods, motivation and theory. SIAM Rev. 19, 46–89.

    Article  Google Scholar 

  6. Golub, G.H. and Wilkinson, J.H., (1976) Ill-conditioned eigensystems and the computation of the Jordan canonical form. SIAM Rev. 18, p. 578–619.

    Article  Google Scholar 

  7. Heinemann, R.F. and Poore, A.B., (1981) Multiplicity, Stability and Oscillatory Dynamics of the Tubulär Reactor, Chem. Eng. Sei. 36, pp. 1411–1419.

    Article  Google Scholar 

  8. Jepson, A. and Spence, A., (1982) Folds in solutions of two para-meter systems and their calculation: Part I. Stanford University Technical Report, (submitted to SIAM JNA).

    Google Scholar 

  9. Jepson, A. and Spence, A.f Paths of Singular Points and their Computation (this proceedings).

    Google Scholar 

  10. Jepson A. and Spence, A., (1983) The numerical Solution of non-linear equations having several parameters,Part I: Scalar Equations. (submitted).

    Google Scholar 

  11. Keller, H.B., Singular Systems, Inverse Iterations and least squares (private communication).

    Google Scholar 

  12. Keller, H.B. and Szeto, R.K-H., (1980) Calculation of flows between rotating disks, in “Computing Methods in Applied Sciences and Engineering” ed. R. Glowinski and J.L. Lions, North Holland, p. 51–61.

    Google Scholar 

  13. Melhem, R.G. and Rheinboldt, W.C. (1982) A comparison of methods for determining turning points of nonlinear equations, Computing, 29, p. 201–226.

    Article  Google Scholar 

  14. Rheinboldt, W.G. and Burkardt, J.V., (1983) “A locally parameterized continuation process”, ACM TOMS, 9, p. 215–235.

    Google Scholar 

  15. Spence, A. and Werner, B. (1982) Nonsimple turning points and cusps, IMA J. of Numer. Anal. 2, p.413–427.

    Google Scholar 

  16. Uppal, A., Ray, W.H. and Poore of continuously stirred tank reactors, Chem. Eng. Sei., 29, p. 967–985.

    Google Scholar 

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Spence, A., Jepson, A.D. (1984). The Numerical Calculation of Cusps, Bifurcation Points and Isola Formation Points in Two Parameter Problems. In: Küpper, T., Mittelmann, H.D., Weber, H. (eds) Numerical Methods for Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 70. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6256-1_35

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  • DOI: https://doi.org/10.1007/978-3-0348-6256-1_35

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-6257-8

  • Online ISBN: 978-3-0348-6256-1

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